Question

1. compare $y = 3^x$ and $y = 7^x$.

a. what is the y intercept of
$y = 3^x$
$y = 7^x$
b. which is steeper?

Explanation:

Part a

For \( y = 3^x \)

Step1: Recall y-intercept definition

The y-intercept of a function is the value of \( y \) when \( x = 0 \).

Step2: Substitute \( x = 0 \) into \( y = 3^x \)

Substitute \( x = 0 \) into the function: \( y=3^{0}\).
We know that any non - zero number to the power of 0 is 1, so \( 3^{0}=1 \). So the y - intercept of \( y = 3^x \) is 1.

For \( y = 7^x \)

Step1: Recall y-intercept definition

The y-intercept of a function is the value of \( y \) when \( x = 0 \).

Step2: Substitute \( x = 0 \) into \( y = 7^x \)

Substitute \( x = 0 \) into the function: \( y = 7^{0}\).
Since any non - zero number to the power of 0 is 1, \( 7^{0}=1 \). So the y - intercept of \( y = 7^x \) is 1.

Part b

Step1: Recall the behavior of exponential functions

For an exponential function of the form \( y = a^{x}\), where \( a>1 \), as the value of \( a \) increases, the graph of the function becomes steeper (grows faster) for \( x>0 \) and approaches 0 faster for \( x < 0 \).

Step2: Compare the bases

We have two exponential functions \( y = 3^{x}\) with base \( a_1=3 \) and \( y = 7^{x}\) with base \( a_2 = 7 \).
Since \( 7>3 \), the function \( y = 7^{x}\) has a steeper graph than \( y=3^{x}\).

Part a Answers

The y - intercept of \( y = 3^x \) is \( \boldsymbol{1} \), and the y - intercept of \( y = 7^x \) is \( \boldsymbol{1} \).

Part b Answer

\( y = 7^{x} \) is steeper.

Answer:

Part a

For \( y = 3^x \)

Step1: Recall y-intercept definition

The y-intercept of a function is the value of \( y \) when \( x = 0 \).

Step2: Substitute \( x = 0 \) into \( y = 3^x \)

Substitute \( x = 0 \) into the function: \( y=3^{0}\).
We know that any non - zero number to the power of 0 is 1, so \( 3^{0}=1 \). So the y - intercept of \( y = 3^x \) is 1.

For \( y = 7^x \)

Step1: Recall y-intercept definition

The y-intercept of a function is the value of \( y \) when \( x = 0 \).

Step2: Substitute \( x = 0 \) into \( y = 7^x \)

Substitute \( x = 0 \) into the function: \( y = 7^{0}\).
Since any non - zero number to the power of 0 is 1, \( 7^{0}=1 \). So the y - intercept of \( y = 7^x \) is 1.

Part b

Step1: Recall the behavior of exponential functions

For an exponential function of the form \( y = a^{x}\), where \( a>1 \), as the value of \( a \) increases, the graph of the function becomes steeper (grows faster) for \( x>0 \) and approaches 0 faster for \( x < 0 \).

Step2: Compare the bases

We have two exponential functions \( y = 3^{x}\) with base \( a_1=3 \) and \( y = 7^{x}\) with base \( a_2 = 7 \).
Since \( 7>3 \), the function \( y = 7^{x}\) has a steeper graph than \( y=3^{x}\).

Part a Answers

The y - intercept of \( y = 3^x \) is \( \boldsymbol{1} \), and the y - intercept of \( y = 7^x \) is \( \boldsymbol{1} \).

Part b Answer

\( y = 7^{x} \) is steeper.